- 8. The Essence of Analysis
- Princeton University Press
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C H A P T E R 8 THE ESSENCE OF ANALYSIS* ANALYSIS is one of three main divisions of mathematics, the other two being (i) Geometry and Topology and (ii) Algebra and Arithmetic. In extent, Analysis is the largest; it comprises subdivisions which are nearly autonomous and which are easier to describe than the whole division. I. MEANING OF ANALYSIS 1. RELATION TO SCIENCE Greek mathematics had a geometrical tenor, as the works of Euclid, Archimedes, Apollonius of Perga, and Pappus testify. It did initiate some durable topics of analysis, but the organized creation of analysis began only in "modern" times around A.D. 1600. Analysis then came into being in stages, growing up in intimacy with Mechanics and Theoretical Physics. This, is not to say that other mathematics does not enter into science too, in fact, all mathematics does. The two great innovations in physics in the 20th century, Relativity and Quantum Theory, had to rely heavily on existing mathematical tools of geometric and algebraic provenance. But whenever basic physics instigated a topic in mathematics spontaneously it was largely in analysis. Thus, differential and integral calculus, ordinary differential equations, and calculus of variations have arisen from mechanics; Fourier series from acoustics and thermodynamics; complex analysis from acoustics, hydrodynamics , and electricity; partial differential equations from elasticity, hydrodynamics, and electrodynamics; and even mathematical probability, which falls under analysis, * Reprinted from Encyclopaedia Britannica, 1961-. Slightly revised ; originally entitled "Analysis." [275] T H E E S S E N C E OF A N A L Y S I S although born from problems of gambling and human chance, drew much of its syllogistic strength in the 19th century from statistical theories of mechanics and thermodynamics . 2. DERIVATIVES Geometry deals with spaces and configurations, topology with spatial deformations, algebra with the general nature of the basic operations of addition, subtraction, multiplication , and division, and arithmetic with additive and multiplicative properties of general "integers." Now, analysis also deals with specific operations, namely with differentiation and integration. But it is more appropriate to say that it deals with the mathematical "infinite" in many of its aspects, as: infinite multitude, infinitely large, infinitely small, infinitely near, infinitely subdivisible, etc. Its first objects and concepts—introductory and yet actively basic —are: infinite sequence, infinite series, a function y — /00 > continuity of a function, derivative of a function Y — df/dx, and integral of a function. (The words "derivative ," "function," "integral" in the English language all appeared in the 16th century; as names in Analysis, "derivative" goes back to G. W. Leibniz in 1676 and "function" in 1692, and "integral" to J. Bernoulli in 1690.) The mathematical concept of derivative is a master concept, one of the most creative concepts in analysis and also in human cognition altogether. Without it there would be no velocity or acceleration or momentum, no density of mass or electric charge or any other density, no gradient of a potential and hence no concept of potential in any part of physics, no wave equation; no mechanics, no physics, no technology, nothing. The formal textbook definition of the concept took over 150 years to evolve, but even to the untutored it will be rewarding to savor Isaac Newton's own description of it under the name of "ultimate ratio" (we quote from page l276] THE ESSENCE OF ANALYSIS 39 of Florian Cajori, Sir Isaac Newton's Mathematical Principles, University of California Press, 1934). For those ultimate ratios with which quantities vanish are not truly the ratios of ultimate quantities, but limits towards which the ratios of quantities, decreasing without a limit, do always converge: and to which they approach nearer than by any given difference, but never go beyond, nor in effect attain to, until the quantities have diminished in infinitum. Newton's generation was literate enough to listen to a recondite mathematical definition, all in words, without symbols; and the Marquise du Chatelet, woman of the world, could undertake to translate Newton's formidable Principia from Latin into French, ultimate ratios and all. The semantic structure of the quoted sentence was well within reach of Greek natural philosophy. Similar sentences can be found in Euclid and Archimedes, and, outside of mathematics, as early as Thucydides. Now, if the inspiration of Greek thinking had been such as to be able to bring about the cognitive content of this sentence as well, then undoubtedly Aristotle would have devoted much of his "metaphysics" to it or even written a special treatise "On the Art of...

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